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http://functions.wolfram.com/04.13.19.0006.01
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Sign[BernoulliB[n]] == (-1)^(n/2 - 1) (1 - n + 2 Floor[n/2]) +
2 KroneckerDelta[n] - KroneckerDelta[n - 1]
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Cell[BoxData[RowBox[List[RowBox[List["Sign", "[", RowBox[List["BernoulliB", "[", "n", "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[FractionBox["n", "2"], "-", "1"]]], RowBox[List["(", RowBox[List["1", "-", "n", "+", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], ")"]]]], "+", RowBox[List["2", RowBox[List["KroneckerDelta", "[", "n", "]"]]]], "-", RowBox[List["KroneckerDelta", "[", RowBox[List["n", "-", "1"]], "]"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation> </semantics> <mi> n </mi> </msub> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> n </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mi> n </mi> </msub> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Sign </ci> <apply> <ci> BernoulliB </ci> <ci> n </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> KroneckerDelta </ci> <ci> n </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sign", "[", RowBox[List["BernoulliB", "[", "n_", "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[FractionBox["n", "2"], "-", "1"]]], " ", RowBox[List["(", RowBox[List["1", "-", "n", "+", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["KroneckerDelta", "[", "n", "]"]]]], "-", RowBox[List["KroneckerDelta", "[", RowBox[List["n", "-", "1"]], "]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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